1. Field of the Invention
The invention relates to a method for calculating power flow solution of a power transmission network, more particularly to a method for calculating power flow solution of a power transmission network that includes a single or multiple unified power flow controllers.
2. Description of the Related Art
Referring to FIG. 1, a power transmission network interface of a typical power system for two areas is shown to include a sending-end bus 811, a demand-end bus 812, a first transmission line 821, a second transmission line 822, a plurality of transmission lines 823, a plurality of power generator buses 814, a plurality of load buses 815, a plurality of power generating devices 83, and a plurality of load devices 84.
The transmission lines 821–823 in general have specific power transmission capacities for transmitting active power and reactive power. To ensure safe transmission of power, the power flow transmitted by these transmission lines must satisfy the following constraint of power transmission capacity:√{square root over ((ActivePower)2+(ReactivePower)2)}{square root over ((ActivePower)2+(ReactivePower)2)}≦transmission line capacity
The first transmission line 821 and the second transmission line 822 are connected electrically to the sending-end bus 811 in area A and the demand-end bus 812 in area B.
The power generating devices 83 are, for instance, power generating plants or other equipment capable of supplying electric power. The load devices 84 are, for instance, residences, factories, etc., in cites or towns. The buses 811–815 are, for instance, substations capable of voltage transformation.
The power generating devices 83 transmit electric power to the load devices 84 through the transmission lines 821–823 and the buses 811–815. During the process of transmission, since the transmission lines 821–823 have inductive and capacitive impedances, the active and reactive powers outputted by the power generating devices 83 are partly lost on the transmission lines 821–823. The remaining active and reactive powers are delivered to the load devices 84 to meet the power demands of residences and factories.
To facilitate description herein, it is assumed that the total power generation is larger than the total load in area A, and that the total power generation is smaller than the total load in area B. Thus, surplus power in area A will be transmitted to area B through the first transmission line 821 and the second transmission line 822 which interconnect the two areas. When the cities or towns in area B grow bigger or there are more factories, the load devices 84 in area B will have a higher demand for power. Then, the power generating devices 83 in area A need to increase their power output, and the flow of active power and reactive power transmitted through the first transmission line 821 and the second transmission line 822 will in turn increase. If the power transmission capacities of the first transmission line 821 and the second transmission line 822 are different, the power flow will be distributed according to the impedance characteristics of the two transmission lines 821, 822, thereby resulting in unbalanced loading of the first and second transmission lines.
For example, supposing the first and second transmission lines 821, 822 have maximum power flow capacity constraints of 200 MVA and 400 MVA, respectively, if area A needs to transmit a total power of 370 MVA to area B, the power flows of the first and second transmission lines 821, 822 will be distributed according to the impedances of the first and second transmission lines 821, 822, which may cause the power flows of the first transmission line 821 and the second transmission line 822 to reach 180 MVA and 190 MVA, respectively. At this time, the power flow of the first transmission line 821 has approached its power flow capacity constraint of 200 MVA, whereas the power flow of the second transmission line 822 is still far below its power flow capacity constraint of 400 MVA. Such a situation of unbalanced loading of the first and second transmission lines 821, 822 will reduce the reliability of the power transmission network.
One way to improve the above situation is to replace the first transmission line 821 with one having a higher power transmission capacity so as to solve the problem of overload associated with the old first transmission line 821. However, such an approach is difficult to implement in practice since the first transmission line 821 may be as long as tens or hundreds of miles. Therefore, re-building the first transmission line 821 will be a huge project.
Referring to FIGS. 2 and 3, another conventional approach is to install a unified power flow controller (UPFC) 7 in the power transmission network and to use the UPFC 7 to regulate the power flow of the first transmission line 821 so as to reduce the loading of the first transmission line 821. It is noted that a receiving-end bus 813 will be added to the power transmission network after installing the UPFC 7 in the power transmission network.
The UPFC 7 is embedded between the sending-end and receiving-end buses 811, 813, and the first transmission line 821 is connected electrically to the receiving-end bus 813 and the demand-end bus 812. The UPFC 7 has the ability to regulate the active and reactive powers transmitted from the UPFC 7 to the receiving-end bus 813 such that the active and reactive power flows transmitted to the demand-end bus 812 through the first transmission line 821 is controllable. In addition, the UPFC 7 can also control the magnitude of voltage on the sending end bus 811 to enhance the voltage stability of the power transmission network.
The UPFC 7 includes a series transformer 71, a series converter 72, a direct current coupling capacitor 73, a shunt converter 74, and a shunt transformer 75, as best shown in FIG. 3.
The direct current coupling capacitor 73 stores a DC voltage with a voltage magnitude of Vdc, and is connected electrically to the series converter 72 and the shunt converter 74. The series converter 72 is connected electrically to the shunt converter 74 and the series transformer 71. The shunt converter 74 is connected electrically to the shunt transformer 75.
The shunt transformer 75 is connected electrically to the sending-end bus 811, and outputs an AC voltage Vs. The series transformer 71 is connected electrically to the sending-end and receiving-end buses 811, 813, and outputs a controllable AC voltage V2 between the sending-end and receiving-end buses 811, 813.
The series transformer 71 controls the active and reactive power flows transmitted via the first transmission line 821 by injecting the AC voltage V2. The shunt transformer 75 maintains the voltage on the sending-end bus 811 stably at a target value Vsref by injecting the AC voltage Vs.
Each of the converters of the UPFC 7 can independently operate as an inverter or a rectifier. When the active power is transmitted from the AC side of the converter to the DC side, the converter is operated as a rectifier. When the active power is transmitted from the DC side of the converter to the AC side, the converter is operated as an inverter.
The shunt converter 74 can absorb or supply active power from or to the sending-end bus 811. The portion of the active power that is absorbed from the sending-end bus 811 is converted to direct current through the shunt transformer 75 and the shunt converter 74 so as to regulate the power stored in the direct current coupling capacitor 73 such that the voltage magnitude on the direct current coupling capacitor 73 can be maintained constant. At the same time, active power is concurrently supplied to the series converter 72.
The series converter 72 converts the power stored in the direct current coupling capacitor 73 to result in the AC voltage V2 of the series transformer 71 that is injected between the sending-end bus 811 and the receiving-end bus 813, and controls the voltage magnitude |V2| and phase angle θ2 of the AC voltage, so as to regulate the active and reactive powers transmitted to the receiving-end bus 813. The transmission of active power between the converters may be in a direction from the shunt converter 74 to the series converter 72, or in a reverse direction.
It is noted that, since the first transmission line 821 is connected to the receiving-end bus 813, the active and reactive powers injected into the first transmission line 821 are the active and reactive powers delivered to the receiving-end bus 813 by the UPFC 7. Therefore, by controlling the power flow delivered to the receiving-end bus 813, the UPFC 7 enables the power flows on the first and second transmission lines 821, 822 to reach a balanced loading condition, and enhances the power reliability of the power transmission network.
In order to adjust the power flow injected into the receiving-end bus 813 to a predetermined control target value, referring to FIGS. 2 to 6, a conventional approach to calculate the power flow solution of a power transmission network that includes a unified power flow controller is disclosed in a paper entitled “Unified power flow controller: a critical comparison of Newton-Raphson UPFC algorithms in power flow studies” by C. R. Fuerte-Esquivel and E. Acha in IEE Proc. Generation, Transmission & Distribution, 1997, and in a paper entitled “A comprehensive Newton-Raphson UPFC model for the quadratic power flow solution of practical power network” by C. R. Fuerte-Esquivel, E. Acha and H. Ambriz-Perez in IEEE Trans. Power System, 2000.
Hereinafter, to facilitate description, the definitions of terms as used herein, such as power flow equation, mismatch vector, etc., are set forth as follows:
A power flow equation is used to describe the balance relationship between the active and reactive powers at each bus in a power transmission network. A power flow equation without considering the power flow controller is explained in detail in the book “Power System Analysis” by A. B. Bergen and V. Vittal, Prentice Hall, 2000.
Take a power transmission network having m buses as an example. A power flow equation without considering the power flow controller can be expressed as follows:Pi−(PGi−PLi)=0 i=1,2, . . . m  equation (1-1)Qi−(QGi−QLi)=0 i=1,2, . . . m  equation (1-2)where PGi and QGi are respectively the active and reactive powers generated by a power generating device connected to a bus i; PLi and QLi are respectively the active and reactive powers consumed by a load connected to the bus i; and Pi and Qi are respectively the total active and reactive powers flowing from the bus i to the transmission lines. Pi and Qi can be expressed as:
                              P          i                =                              ∑                          k              =              1                        n                    ⁢                                                                  V                i                                                    ⁢                                                                            V                  k                                                            ⁡                              [                                                                            G                      ik                                        ⁢                                          cos                      ⁡                                              (                                                                              θ                            i                                                    -                                                      θ                            k                                                                          )                                                                              +                                                            B                      ik                                        ⁢                                          sin                      ⁡                                              (                                                                              θ                            i                                                    -                                                      θ                            k                                                                          )                                                                                            ]                                                                        equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            3                    )                                                  Q          i                =                              ∑                          k              =              1                        n                    ⁢                                                                  V                i                                                    ⁢                                                                            V                  k                                                            ⁡                              [                                                                            G                      ik                                        ⁢                                          sin                      ⁡                                              (                                                                              θ                            i                                                    -                                                      θ                            k                                                                          )                                                                              +                                                            B                      ik                                        ⁢                                          cos                      ⁡                                              (                                                                              θ                            i                                                    -                                                      θ                            k                                                                          )                                                                                            ]                                                                        equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            4                    )                    where Gik+jBik is the admittance of the transmission lines, and can be obtained by calculating the reciprocal of the impedances of the transmission lines; n is the number of transmission lines connected to the bus i; |Vi| and |Vk| are voltage magnitudes at bus i and bus k; and θi and θk are phase angles at bus i and bus k.
The active power delivered to a bus must be equal to the active power delivered from the bus, and the reactive power delivered to a bus must be equal to the reactive power delivered from the bus. By grouping together the equations of the buses, a power flow equation of the power system can be obtained.
Elements of a mismatch vector include the amounts of imbalance of the active power and the reactive power on all the buses in the transmission network in each iteration. Each active power and each reactive power are functions of voltage magnitudes and phase angles at the buses, respectively. Therefore, the mismatch vector can be expressed as:
                                          f            ⁡                          (                              x                                  (                  k                  )                                            )                                =                                                    [                                                                                                    P                        i                                            ⁡                                              (                                                  x                                                      (                            k                            )                                                                          )                                                              -                                          (                                                                        P                          Gi                                                -                                                  P                          Li                                                                    )                                                                                                                          Q                        i                                            ⁡                                              (                                                  x                                                      (                            k                            )                                                                          )                                                              -                                          (                                                                        Q                          Gi                                                -                                                  Q                          Li                                                                    )                                                                      ]                            ⁢                                                          ⁢              i                        =            1                          ,        2        ,        …        ⁢                                  ,        m                            equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            5                    )                    where x(k) is the kth iteration value of state vector x. The state vector x is composed of the voltage magnitudes and phase angles at the buses, and can be expressed as:
                              x          =                                                    [                                                      θ                    i                                                                                                  V                      i                                                                                          ]                            ⁢                                                          ⁢              i                        =            1                          ,        2        ,        …        ⁢                                  ,        m                            equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            6                    )                    
As shown in FIG. 4, the approach proposed by C. R. Fuerte-Esquivel et al. is to adopt a voltage source-based static model to represent the UPFC 7. The static model employs a series voltage source 61 and a series impedance 62 having an impedance value of Rser+jXser to model the series converter 72 and the series transformer 71. The series voltage source 61 outputs an AC voltage Vser. The AC voltage Vser has a voltage magnitude |Vser| and a phase angle θser. The series impedance 62 represents the impedance of the series transformer 71.
The shunt converter 74 and the shunt transformer 75 are modeled by a shunt voltage source 63 with a voltage Vsh, and a shunt impedance 64 with an impedance value of Rsh+jXsh. The voltage magnitude and phase angle of the shunt voltage source 63 are |Vsh| and θsh, respectively. Similarly, the shunt impedance 64 represents the impedance of the shunt transformer 75.
The static model relies on the voltage sources and impedances to result in an effect equivalent to that attributed to the UPFC 7 on the power flow between the sending-end and receiving-end buses 811, 813. The impedances Rser, Rsh, Xser and Xsh can be obtained from specifications provided by manufacturers of the transformers, whereas the output voltage magnitudes |Vsh|, |Vser| and phase angles θsh, θser of the voltage sources are control variables of the UPFC 7. The control variables can be controlled independently through pulse width modulation (PWM) in the converters. The voltage sources and impedances in the static model are used to estimate equivalent loads Pr, Qr, Ps, and Qs caused by the UPFC 7 on the sending-end and receiving-end buses 811, 813.
Referring to FIGS. 2 to 6, the conventional approach uses the static model in conjunction with the Newton-Raphson algorithms to calculate solutions of |Vsh|, θsh, |Vser|, and θser. The calculating method includes the following steps:
In step 901, an initial vector value x(k) is given to the state vector x, the elements in the state vector x including control variables |Vsh|, θsh, |Vser|, and θser relevant to the UPFC 7, and the voltage magnitude |Vi| and phase angle θi at each bus in the power transmission network.
However, since a power transmission network generally includes a large number of buses, to facilitate description hereinafter, only the voltage magnitudes and phase angles at those buses that are connected to the UPFC 7 are provided when the state vector is described. In the following, it is assumed that the voltage magnitude and phase angle at the sending-end bus 811 are |Vs| and θs, respectively, and that the voltage magnitude and the phase angle at the receiving-end bus 813 are |Vr| and θr, respectively.
In this step, using the power transmission network illustrated in FIG. 2 as an example, the elements in the state vector x include the voltage magnitude |Vs| and phase angle θs at the sending-end bus 811, and the voltage magnitude |Vr| and phase angle θr at the receiving-end bus 813. Therefore, after executing step 901, the initial vector value x(k) of the state vector x includes |Vs|(k), θs(k), |Vr|(k), θr(k), |Vsh|(k), θsh(k), |Vser|(k), and θser(k).
In step 902, the state vector value x(k) is substituted into the mismatch vector f of the power transmission network shown in FIG. 6 to calculate the value f(x(k)) of the mismatch vector f while neglecting the equivalent load of the UPFC 7 on the power transmission network.
It is noted that, in the power transmission network shown in FIG. 6, although the UPFC 7 of FIG. 2 is removed, it is still necessary to retain the receiving-end bus 813. The mismatch vector f is as shown in equation (1-5).
In step 903, the state vector x(k) is substituted into a Jacobian matrix J of the power transmission network of FIG. 6 to calculate the value J(x(k)) of the Jacobian matrix J while neglecting the equivalent load of the UPFC 7 on the power transmission network.
The Jacobian matrix J is the first-order partial derivatives of the mismatch vector f with respect to the state vector x.
In step 904, equations (1-7) and (1-8) are used to calculate the equivalent load caused by the UPFC 7 on the receiving-end bus 813, i.e., the active and reactive powers Pr, Qr absorbed by the UPFC 7 from the receiving-end bus 813 are expressed in terms of |Vs|, θs, |Vr|, θr, |Vsh|, θsh, |Vser|, and θser of the state vector x. The values of |Vs|(k), θs(k), |Vr|(k), θr(k), |Vsh|(k), θsh(k), |Vser|(k), and θser(k) are also substituted into Pr, Qr to obtain Pr(x(k)), Qr(x(k)).
                              P          r                =                                                                                              V                  r                                                            2                        ⁢                          G              rr                                +                                                                  V                s                                                    ⁢                                                        V                r                                                    ⁢                          (                                                                    G                    rs                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          s                                                                    )                                                                      +                                                      B                    rs                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          s                                                                    )                                                                                  )                                +                                                                  V                r                                                    ⁢                                                        V                ser                                                    ⁢                          (                                                                    G                    rr                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          ser                                                                    )                                                                      +                                                      B                    rr                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          ser                                                                    )                                                                                  )                                                          equation        ⁢                                  ⁢                  (                      1            -            7                    )                    
                              Q          r                =                                            -                                                                                      V                    r                                                                    2                                      ⁢                          B              rr                                +                                                                  V                s                                                    ⁢                                                        V                r                                                    ⁢                          (                                                                    G                    rs                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          s                                                                    )                                                                      -                                                      B                    rs                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          s                                                                    )                                                                                  )                                +                                                                  V                r                                                    ⁢                                                        V                ser                                                    ⁢                          (                                                                    G                    rr                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          ser                                                                    )                                                                      -                                                      B                    rr                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          r                                                -                                                  θ                          ser                                                                    )                                                                                  )                                                          equation        ⁢                                  ⁢                  (                      1            -            8                    )                    where Grr, Brr, Grs, and Brs can be obtained from the following equations:
                    G        rr            +              jB        rr              =          1                        R          ser                +                  jX          ser                                        G        rs            +              jB        rs              =          -              1                              R            ser                    +                      jX            ser                              
In step 905, equations (1-9) and (1-10) are used to calculate the equivalent load caused by the UPFC 7 on the sending-end bus 811, i.e., the active and reactive powers Ps, Qs absorbed by the UPFC 7 from the sending-end bus 811 are expressed in terms of |Vs|, θs, |Vr|, θr, |Vsh|, θsh, |Vser|, and θser. The values of |Vs|(k), θs(k), |Vr|(k), θr(k), |Vsh|(k), θsh(k), |Vser|(k), and θser(k) are substituted into Ps, Qsto obtain the values of Ps(x(k)), Qs(x(k)).
                                                                        P                s                            =                            ⁢                                                                                                                                        V                        s                                                                                    2                                    ⁢                                      G                    ss                                                  +                                                                                                V                      s                                                                            ⁢                                                                                V                      r                                                                            ⁢                                      (                                                                                            G                          sr                                                ⁢                                                  cos                          ⁡                                                      (                                                                                          θ                                s                                                            -                                                              θ                                r                                                                                      )                                                                                              +                                                                        B                                                                                                                                            ⁢                            sr                                                                          ⁢                                                  sin                          ⁡                                                      (                                                                                          θ                                                                                                                                                                          ⁢                                  s                                                                                            -                                                              θ                                                                                                                                                                          ⁢                                  r                                                                                                                      )                                                                                                                )                                                  +                                                                                                      ⁢                                                                                                              V                      s                                                                            ⁢                                                                                V                      ser                                                                            ⁢                                      (                                                                                            G                          sr                                                ⁢                                                  cos                          ⁡                                                      (                                                                                          θ                                s                                                            -                                                              θ                                ser                                                                                      )                                                                                              +                                                                        B                          sr                                                ⁢                                                  sin                          ⁡                                                      (                                                                                          θ                                s                                                            -                                                              θ                                ser                                                                                      )                                                                                                                )                                                  +                                                                                                      ⁢                                                                                      V                    s                                                                    ⁢                                                                        V                    sh                                                                    ⁢                                  (                                                                                    G                        sh                                            ⁢                                              cos                        ⁡                                                  (                                                                                    θ                              s                                                        -                                                          θ                              sh                                                                                )                                                                                      +                                                                  B                        sh                                            ⁢                      sin                      ⁢                                              (                                                                              θ                            s                                                    -                                                      θ                            sh                                                                          )                                                                              )                                                                                        equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            9                    )                    
                                                                        Q                s                            =                            ⁢                                                                    -                                                                                                                    V                          s                                                                                            2                                                        ⁢                                      B                    ss                                                  +                                                                                                V                      s                                                                            ⁢                                                                                V                      r                                                                                                                                                                                ⁢                                                (                                                                                    G                        sr                                            ⁢                                              sin                        ⁡                                                  (                                                                                    θ                              s                                                        -                                                          θ                              r                                                                                )                                                                                      +                                                                  B                        sr                                            ⁢                                              cos                        ⁡                                                  (                                                                                    θ                              s                                                        -                                                          θ                              r                                                                                )                                                                                                      )                                +                                                                                                      ⁢                                                                                                              V                      s                                                                            ⁢                                                                                V                      ser                                                                            ⁢                                      (                                                                                            G                          sr                                                ⁢                                                  sin                          ⁡                                                      (                                                                                          θ                                s                                                            -                                                              θ                                ser                                                                                      )                                                                                              -                                                                        B                          sr                                                ⁢                                                  cos                          ⁡                                                      (                                                                                          θ                                s                                                            -                                                              θ                                ser                                                                                      )                                                                                                                )                                                  +                                                                                                      ⁢                                                                                      V                    s                                                                    ⁢                                                                        V                    sh                                                                    ⁢                                  (                                                                                    G                        sh                                            ⁢                                              sin                        ⁡                                                  (                                                                                    θ                              s                                                        -                                                          θ                              sh                                                                                )                                                                                      -                                                                  B                        sh                                            ⁢                                              cos                        ⁡                                                  (                                                                                    θ                              s                                                        -                                                          θ                              sh                                                                                )                                                                                                      )                                                                                        equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            10                    )                    where Gss, Bss, Gsr, Bsr, Gsh and Bsh can be obtained from the following equations:
                    G        ss            +              jB        ss              =                  -                  1                                    R              ser                        +                          jX              ser                                          +              1                              R            sh                    +                      jX            sh                                                  G        sr            +              jB        sr              =          -              1                              R            ser                    +                      jX            ser                                                  G        sh            +              jB        sh              =          -              1                              R            sh                    +                      jX            sh                              
In step 906, the sum
      P    r    +      P    line    ref  of the equivalent active load Pr of the UPFC 7 at the receiving-end bus 813 and the control target value
      P    line    ref    ,the sum
      Q    r    +      Q    line    ref  of the equivalent reactive load Qr of the UPFC 7 at the receiving-end bus 813 and the control target value
      Q    line    ref    ,and the difference |Vs|−Vsref between the voltage |Vs| at the sending-end bus 811 and the control target value Vsref are calculated.
In step 907, the mismatch of an active power balance equation Pdc is calculated based on equation (1-11):Pdc=Pser+Psh  equation (1-11)where Pser is the active power that the series converter 72 injected into the AC side, and can be obtained based on equation (1-12):
                              P          ser                =                ⁢                                                                                              V                  ser                                                            2                        ⁢                          G              rr                                +                    ⁢                                                                  V                ser                                                    ⁢                                                        V                s                                                    ⁢                          (                                                                    G                    sr                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          ser                                                -                                                  θ                          s                                                                    )                                                                      +                                                      B                    sr                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          ser                                                -                                                  θ                          s                                                                    )                                                                                  )                                +                    ⁢                                                                  V                ser                                                    ⁢                                                        V                r                                                    ⁢                          (                                                                    G                    rr                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          ser                                                -                                                  θ                          r                                                                    )                                                                      +                                                      B                    rr                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          ser                                                -                                                  θ                          r                                                                    )                                                                                  )                                                          equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            12                    )                    and Psh is the active power that the shunt converter 74 injected into the AC side, and can be obtained from equation (1-13):
                              P          sh                =                                            -                                                                V                  sh                                                                      ⁢                          G              sh                                +                                                                  V                sh                                                    ⁢                                                        V                s                                                    ⁢                          (                                                                    G                    sh                                    ⁢                                      cos                    ⁡                                          (                                                                        θ                          sh                                                -                                                  θ                          s                                                                    )                                                                      +                                                      B                    sh                                    ⁢                                      sin                    ⁡                                          (                                                                        θ                          sh                                                -                                                  θ                          s                                                                    )                                                                                  )                                                          equation        ⁢                                  ⁢                  (                      1            -            13                    )                    
In step 908, the value of the mismatch vector f′ of the power transmission network that includes the UPFC 7 is modified based on equation (1-14):f′=f+ΔfUPFC  equation (1-14)where f is the value of the mismatch vector while neglecting the UPFC 7, and was obtained in step 902; and ΔfUPFC is the amount of adjustment added as a result of inclusion of the UPFC 7, and is calculated based on equation (1-15):
                              Δ          ⁢                                          ⁢                      f            UPFC                          =                  [                                                                      P                  s                                                                                                      Q                  s                                                                                                      P                  r                                                                                                      Q                  r                                                                                                                          P                    r                                    +                                      P                    r                    ref                                                                                                                                            Q                    r                                    +                                      Q                    r                    ref                                                                                                                                            V                    s                                    -                                      V                    s                    ref                                                                                                                        P                                      d                    ⁢                                                                                  ⁢                    c                                                                                ]                                    equation        ⁢                                  ⁢                  (                      1            ⁢                          -                        ⁢            15                    )                    
In step 909, the value J′ of the Jacobian matrix of the power transmission network that includes the UPFC 7 is modified based on equation (1-16):J′(x(k))=J(x(k))+∂ΔfUPFC/∂x  equation (1-16)where J(x(k)) is the value of the Jacobian matrix while neglecting the UPFC 7, and was obtained in step 903; and ∂ΔfUPFC/∂x is the amount of adjustment.
In step 910, f′(x(k)) and J′(x(k)) calculated in steps 908 and 909 are substituted into the following equation to calculate the updated state vector x(k+1):x(k+1)=x(k)−J′−1(x(k))f′(x(k))  equation (1-17)
In step 911, it is determined whether f′(x(k)) is smaller than an tolerable error. If f′(x(k)) is smaller than the tolerable error, this indicates that the state vector x has converged to the solution of the power flow equation. Conversely, x(k+1) is used as an updated x(k), and the flow skips back to step 902. Steps 902 to 911 are repeated until f′(x(k)) is smaller than the tolerable error.
In step 912, after the state vector x has converged, the converters 72, 74 are set with |Vsh|, θsh, |Vser|, and θser of the state vector x using, for example, PWM techniques. At this time, the power flow of the first transmission line 821 will achieve the predetermined control target values
  P  line  refand
      Q    line    ref    ,and the voltage magnitude at the sending-end bus 811 will also achieve the predetermined control target Vsref.
However, when the converters 72, 74 convert the power flow between alternating currents and direct currents, due to the conversion efficiency of the converters 72, 74, the active power of the direct current coupling capacitor 73 cannot be 100% converted into the buses 811, 813, and a part of the power will be lost in the converters 72, 74.
Although the conventional approach has considered the active power consumed by the transformers 71, 75 when calculating the active power balance equations in equations (1-12) and (1-13), it fails to consider the loss attributed to the converters 72, 74, so that although the solution |Vsh|, θsh, |Vser|, and θser of the power flow equation is used to set the converters 72, 74, the power flow transmitted to the receiving-end bus 813 still cannot achieve the control target values
  P  line  refand
  Q  line  refprecisely.
Furthermore, the aforesaid conventional approach can only be adapted for use when the shunt converter 74 and the series converter 72 are operated in specific operating modes. That is, the shunt converter 74 is operated in an automatic voltage control mode to control voltage magnitude of the sending-end bus 811 at a fixed value, whereas the series converter 72 is operated in an automatic flow control mode to simultaneously control the active power and the reactive power transmitted to the receiving-end bus 813.
In addition, discretion has to be exercised when setting an initial value for the state vector x(k) in step 901 of the aforesaid conventional approach. If an improper initial value is selected, the solution of the power flow equation will not converge or the iterative solution of the state variable will oscillate.